Weak Gravity, MOND, and State Vector Reduction

Carl Brannen,
September 25, 2003 -

Abstract
This short note takes Roger Penrose's idea that entropy and quantum
state vector reduction must be associated with gravity, along
with an additional assumption that momentum or velocity be
quantized rather than continuous, and shows that very weak
gravitational fields must be modified in a manner that matches
the MOND gravitational anomaly.

Roger Penrose has written some compelling arguments for
associating gravitational fields with state vector reduction
and entropy. In short, the reasoning goes that Stephen Hawking's
demonstration that black holes are associated with entropy,
along with the fact that black holes violate Liouville's
theorem about the constancy of phase space volume, imply
that entropy must be associated with a quantum gravity
effect that is due to gravity in general, and not just
black holes. But his argument implies that there might also
be an effect in the reverse direction. That is, that
wave function collapse may modify the effective gravitational
field. Furthermore, since it appears to be evident that
entropy is as active in the weak gravitational field limit
as in the strong (where Penrose makes his case) the place
to look for gravitational anomalies due to wave function
collapse is in the weak gravitational limit. But there
is already experimental evidence of something odd going
on with weak gravitational fields due to Mordecai Milgrom's
MOND (MOdified Newtonian Dynamics) explanation for the
gravitational anomalies of galactic rotation data usually
attributed to dark matter. The purpose of this note is
to suggest a connection between these theories.

One of the most intriguing effects of state vector reduction
is the Quantum Zeno Effect, where repeated measurements of
a quantum system are shown to result in a suppression
of the time evolution of the wave function. The effect
is due to the fact that perturbations in quantum mechanics
always show up as the square of the wave function. In
order to calcuate the perturbation, one needs to have
a discrete set of eigenstates, rather than continuous.
In this condition, a perturbational analysis of the
time evolution of a wave state that begins as an
eigenstate, must show that the eigenstate is abandoned
with a time dependency proportional to the square of
time rather than (as exponential decay implies)
simply proportional to time. It can be shown that
as time passes, the perturbational approximation
eventually fails and exponential decay follows. In
short, the effect depends on (a) discrete eigenstates,
and (b) wave function collapse occuring at a rate
faster than the rate at which perturbational analysis
fails.

To assist the calculation, it helps to imagine an
experiment where a test particle is used to calculate
the gravitational acceleration of a spherical massive body.
If we assume that the test particle is negatively
charged, we can apply a negative charge to the massive
body that will balance the gravitational acceleration,
and then from our knowledge of electricity, we can
calculate the acceleration of the massive body.
Furthermore, since the massive body is spherical,
the cancellation of the gravitaional and electrical
force must extend over all space. That is, the
cancellation must occur in both the strong field
regions of the gravitational field and the weak
field regions.

The reason for going about the problem in this way
is because we really don't know how to include
gravity, a curvature related effect, into quantum
mechanics. So instead of beating our heads in
guessing at a form for the interaction, we can
instead use perturbational analysis off of the
states that nature herself provides for an
charged particle caught in a gravitational
potential. Since we don't know the actual wave
function, we cannot calculate the acceleration
directly, but we can calcuate the amount of
electric field required to accelerate the
particle just enough to cancel the gravitational
attraction. Since the electric attraction
is different depending on the ratios of the
charges, the electric calcuation cannot depend
on the curvature of space-time around the
massive body. What is happening is that we
are using electric fields, a situation that
we understand at very high levels of accuracy,
to calibrate (I want to write "gauge") the
gravitational field.

In order for perturbational analysis to apply,
we need to have that the unperturbed particle
is in an eigenstate. Since the gravitational
field is not actually a field, but instead
is a curvature effect, the eigenstates of
the particle (with the massive body uncharged)
will have to correspond to the particle being
attracted to the massive body. It is these
eigenstates that must be perturbed.

The first requirement for the Quantum Zeno
Effect to apply is that the eigenstates be
discrete rather than continuous. Of course
our particle is bound, therefore its eigenstates
must correspond to discrete eigenvalues.
What this means in the context of gravitation
is unclear, as these eigenstates must correspond
to motion towards the gravitating body.
It can also be argued, in a way reminiscent
of Milgrom's reasoning that the hubble acceleration
has something to do with the MOND effect,
that the finite size of the universe prevents
"box normalization" from being taken all the
way to infinity, and that therefore there
must be a quantization of momentum similar
to that seen in a finite box. I think this
last argument is unconvincing, but instead
that momenta, or more particularly, velocities
are quantized for reasons similar to the
quantization of angular momenta, as explained
in my recent article: "Ether, Relativity, Gauges
and Quantum Mechanics", Carl Brannen, September
2003.

The second requirement of the Quantum Zeno
Effect is that the wave function collapse
rate has to be fast enough, compared to the
rate at which perturbation theory fails. This
corresponds to the weak limit for the electric
field, and therefore also for the gravitational
field. So if our test is done sufficiently
far from a sufficiently weak massive body, this
criterion will be satisfied as well. But note
that this requirement will fail close to the
gravitating body, where, therefore, the
usual calculations for electric acceleration
must equal the usual calculations for gravitational
acceleration.

With these two assumptions, the Quantum Zeno
Effect applies, and this indicates that in
the limit of very weak gravitational fields,
the electric field will be able to move the
particle from eigenstate to eigenstate with
a rapidity only proportional to the square
of the field rather than linearly proportional
to the field as Newton or Einstein would imply.
Thus, if we use the gravitational potential
defined by Newton / Einstein, the two fields
will be unable to be cancelled with a single
electric charge on the massive body.

The acceleration of the electric field depends
on the mass and charge of a test particle used
to measure it. Because of this fact, it is not
possible for the universe to correct for the
above discrepancy by making the electric field
weak for certain particles in certain
gravitational fields. Instead, the universe must
maintain its happy equilibrium by making the
gravitational field have an effective strength
that is sufficient to cancel the effects that
the Zeno effect would have in the region of
weak attractions.

Equating the field strength of the gravitational
field with the electric field, yields the MOND
relation, but this discussion does not eliminate
the possiblity that a0 might
depend on the mass of the gravitating body.
For that, either the modified relativity of the
above Brannen article is needed, or perhaps
a better estimate of the bound states based
on an understanding of quantum gravity.